Relative Ding Projective Modules over Formal Triangular Matrix Rings

Let U be a (B, A)-bimodule, A and B be rings, and be a formal triangular matrix ring. In this paper, we characterize the structure of relative Ding projective modules over T under some conditions. Furthermore, using the left global relative Ding projective dimensions of A and B, we estimate the relative Ding projective dimension of a left T-module.

PS-modules over Rings of Generalized Power Series

Let R be a commutative ring and (S, ≤) a strictly totally ordered monoid which satisfies the condition that 0 ≤ s for every s ∈ S. In this paper we show that if RM is a PS-module, then the module [[MS≤]] of generalized power series over M is a PS [[RS,≤]]-module.

F-perfect Rings and Modules

Let R be a ring, and let （F, C） be a cotorsion theory. In this article, the notion of F-perfect rings is introduced as a nontrial generalization of perfect rings and A-perfect rings. A ring R is said to be right dr-perfect if F is projective relative to R for any F ∈ F. We give some characterizations of F-perfect rings. For example, we show that a ring R is right F-perfect if and only if F-covers of finitely generated modules are projective. Moreover, we define F-perfect modules and investigate some properties of them.

PS-injective Modules, PS-flat Modules and PS-coherent Rings

A left ideal I of a ring R is small in case for every proper left ideal K of R, K ＋ I ≠R. A ring R is called left PS-coherent if every principally small left ideal Ra is finitely presented. We develop, in this paper, PS-coherent rings as a generalization of P-coherent rings and J-coherent rings. To characterize PS-coherent rings, we first introduce PS-injective and PS-flat modules, and discuss the relation between them over some spacial rings. Some properties of left PS-coherent rings are also studied.

Some Rings Characterized by Modules

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On FI-extending Rings and Modules

In this paper, we discuss FI-extending property of rings and modules. The main results are the following： a characterization of von Neumann regular rings which are two-sided FI-extending is given; sufficient conditions for direct summands of FI-extending modules to be FI-extending are obtained; and at last, a necessary and sufficient condition for nonsingular modules over nonsingular rings to be FI-extending is given.

<i>S</i>-Injective Modules and Rings

We introduce and investigate the concept of s-injective modules and strongly s-injective modules. New characterizations of SI-rings, GV-rings and pseudo-Frobenius rings are given in terms of s-injectivity of their modules.

Gorenstein projective modules over rings of Morita contexts

Under semi-weak and weak compatibility conditions of bimodules,we establish necessary and sufficient conditions of Gorenstein-projective modules over rings of Morita contexts with one bimodule homomorphism zero.This extends greatly the results on triangular matrix Artin algebras and on Artin algebras of Morita contexts with two bimodule homomorphisms zero in the literature,where only sufficient conditions are given under a strong assumption of compatibility of bimodules.An application is provided to describe Gorenstein-projective modules over noncommutative tensor products arising from Morita contexts.Our results are proved under a general setting of noetherian rings and modules instead of Artin algebras and modules.

L-fuzzy Ring and L-fuzzy Module

Based on LF group theory[1], we introduce the concepts LF ring and LF module and disscuss their properties.

Homological properties of modules characterized by matrices

Some homological properties of R-modules were investigated by matrices over aring R. Given two cardinal numbers α, β and an α x β row-finite matrix A, it was proved thatExt_R^1(R^((α))/R^((β))A, M) = 0 if and only if M_α/r_(M_α)(R^((β))A) ≈ Hom_R(R^((β))A,M) ifand only if r_(M_β)l_(R^((β)))(A) = AM_α. Thus, the notion of (m,n)-injectivity was extended.Moreover, ( α, β) -flatness was characterized via annihilators of matrices, factorizations ofhomomorphisms as well as homological groups so that (m, n)-flat modules, f-projective modules andn-projective modules were consolidated under the notion of (α, β)-flat modules. Furthermore, acharacterization of left R-ML modules and some equivalent conditions for R^((β)) to be left R-MLwere presented. Consequently, the notions of coherent rings, (m, n)-coherent rings and π-coherentrings were consolidated under that of (α, β)-coherent rings.

ABSOLUTELY E-PURE MODULES AND E-PURE SPLIT MODULES

We first introduce the concepts of absolutely E-pure modules and E-pure split modules. Then, we characterize the IF rings in terms of absolutely E-pure modules. The E-pure split modules are also characterized.

Zip Modules

A ring R is called right zip provided that if the annihilator τR（X） of a subset X of R is zero, then τR（Y） = 0 for some finite subset Y C X. Such rings have been studied in literature. For a right R-module M, we introduce the notion of a zip module, which is a generalization of the right zip ring. A number of properties of this sort of modules are established, and the equivalent conditions of the right zip ring R are given. Moreover, the zip properties of matrices and polynomials over a module M are studied.

An Introduction to the Theory of Matrix Near-Rings

Matrix rings are prominent in abstract algebra. In this paper we give an overview of the theory of matrix near-rings. A near-ring differs from a ring in that it does not need to be abelian and one of the distributive laws does not hold in general. We introduce two ways in which matrix near-rings can be defined and discuss the structure of each. One is as given by Beildeman and the other is as defined by Meldrum. Beildeman defined his matrix near-rings as normal arrays under the operation of matrix multiplication and addition. He showed that we have a matrix near-ring over a near-ring if, and only if, it is a ring. In this case it is not possible to obtain a matrix near-ring from a proper near-ring. Later, in 1986, Meldrum and van der Walt defined matrix near-rings over a near-ring as mappings from the direct sum of n copies of the additive group of the near-ring to itself. In this case it can be shown that a proper near-ring is obtained. We prove several properties, introduce some special matrices and show that a matrix notation can be introduced to make calculations easier, provided that n is small.

*-Modules, co-*-modules and cotilting modules over Noetherian rings

Let R be a Noetherian ring. The projectivity and injectivity of modules over R are discussed. The concept of modules is introduced and the descriptions for co-*-modules over R are given. At last, cotilting modules over R are characterized by means of co-*-modules.

Higher Level Orderings on Modules

The aim of this paper is to investigate higher level orderings on modulesover commutative rings. On the basis of the theory of higher level orderings on fields andcommutative rings, some results involving existence of higher level orderings are generalized to thecategory of modules over commutative rings. Moreover, a strict intersection theorem for higherlevel orderings on modules is established.

Minimal silting modules and ring extensions

Ring epimorphisms often induce silting modules and cosilting modules,termed minimal silting or minimal cosilting.The aim of this paper is twofold.Firstly,we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra.In particular,we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand.Secondly,we discuss the behavior of minimality under ring extensions.We show that minimal cosilting modules over a commutative noetherian ring extend to minimal cosilting modules along any flat ring epimorphism.Similar results are obtained for commutative rings of small homological dimensions.

Gorenstein AC-Projective and AC-Injective Modules over Formal Triangular Matrix Rings

Let A and B be rings and U a(B,A)-bimodule.If BU is flat and UA is finitely generated projective(resp.,BU is finitely generated projective and UA is flat),then the characterizations of level modules and Gorenstein AC-projective modules(resp.,absolutely clean modules and Gorenstein AC-injective modules)over the formal triangular matrix ring T=(A0 UB)are given.As applications,it is proved that every Gorenstein AC-projective left T-module is projective if and only if each Gorenstein AC-projective left A-module and B-module is projective,and every Gorenstein AC-injective left T-module is injective if and only if each Gorenstein AC-injective left A-module and B-module is injective.Moreover,Gorenstein AC-projective and AC-injective dimensions over the formal triangular matrix ring T are studied.

EXPLICIT DESCRIPTION OF A CLASS OF INDECOMPOSABLE INJECTIVE MODULES

Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module R/p.

Almost Cotorsion Modules

In this paper, we introduce the concept of almost cotorsion modules. A module is called almost cotorsion if it is subisomorphic to its cotorsion envelope. Some characterizations of almost cotorsion modules are given. It is also proved that every module is a direct summand of an almost cotorsion module. As an application, perfect rings are characterized in terms of almost cotorsion modules.

（Inco） Projective Modules of Relatively Hereditary Torsion Theroy

In the paper, we define（inco） project modules of relatively hereditary torsion theory τ by intersection complement of module and study their properties; secondly, we define the（inco） τ-semisimple ring by（inco） τ-projective module and study their properties. When r is a trivial torsion theory on R-rood, we prove that R is a semisimple ring if and only if R is a（inco） semisimple ring and satisfies（inco） condition.